# 1.2 2d dft

 Page 1 / 1
This module extends the ideas of the Discrete Fourier Transform (DFT) into two-dimensions, which is necessary for any image processing.

## 2d dft

To perform image restoration (and many other useful image processing algorithms) in a computer, we need a FourierTransform (FT) that is discrete and two-dimensional.

$F(k, l)=(u=\frac{2\pi k}{N}, v=\frac{2\pi l}{N}, F(u, v))$
for $k=\{0, , N-1\}$ and $l=\{0, , N-1\}$ .
$F(u, v)=\sum \sum f(m, n)e^{-(ium)}e^{-(ivm)}$
$F(k, l)=\sum_{m=0}^{N-1} \sum_{n=0}^{N-1} f(m, n)e^{-i\frac{2\pi km}{N}}e^{-i\frac{2\pi ln}{N}}$
where the above equation ( ) has finite support for an $N$ x $N$ image.

## Inverse 2d dft

As with our regular fourier transforms, the 2D DFT also has an inverse transform that allows us to reconstruct an imageas a weighted combination of complex sinusoidal basis functions.

$f(m, n)=\frac{1}{N^{2}}\sum_{k=0}^{N-1} \sum_{l=0}^{N-1} F(k, l)e^{\frac{i\times 2\pi km}{N}}e^{\frac{i\times 2\pi ln}{N}}$

## 2d dft and convolution

The regular 2D convolution equation is

$g(m, n)=\sum_{k=0}^{N-1} \sum_{l=0}^{N-1} f(k, l)h(m-k, n-l)$

Below we go through the steps of convolving two two-dimensional arrays. You can think of $f$ as representing an image and $h$ represents a PSF, where $h(m, n)=0$ for $m\land n> 1$ and $m\land n< 0$ . $h=\begin{pmatrix}h(0, 0) & h(0, 1)\\ h(1, 0) & h(1, 1)\\ \end{pmatrix}$ $f=\begin{pmatrix}f(0, 0) & & f(0, N-1)\\ & & \\ f(N-1, 0) & & f(N-1, N-1)\\ \end{pmatrix}$ Step 1 (Flip $h$ ):

$h(-m, -n)=\begin{pmatrix}h(1, 1) & h(1, 0) & 0\\ h(0, 1) & h(0, 0) & 0\\ 0 & 0 & 0\\ \end{pmatrix}$
Step 2 (Convolve):
$g(0, 0)=h(0, 0)f(0, 0)$
We use the standard 2D convolution equation ( ) to find the first element of our convolved image. In order to better understand what ishappening, we can think of this visually. The basic idea is to take $h(-m, -n)$ and place it "on top" of $f(k, l)$ , so that just the bottom-right element, $h(0, 0)$ of $h(-m, -n)$ overlaps with the top-left element, $f(0, 0)$ , of $f(k, l)$ . Then, to get the next element of our convolved image, we slide the flipped matrix, $h(-m, -n)$ , over one element to the right and get the following result: $g(0, 1)=h(0, 0)f(0, 1)+h(0, 1)f(0, 0)$ We continue in this fashion to find all of the elements ofour convolved image, $g(m, n)$ . Using the above method we define the general formula to find a particular element of $g(m, n)$ as:
$g(m, n)=h(0, 0)f(m, n)+h(0, 1)f(m, n-1)+h(1, 0)f(m-1, n)+h(1, 1)f(m-1, n-1)$

## Circular convolution

What does $H(k, l)F(k, l)$ produce?

2D Circular Convolution

$\stackrel{~}{g}(m, n)=\mathrm{IDFT}(H(k, l)F(k, l))=\mathrm{circularconvolutionin2D}$

Due to periodic extension by DFT ( ):

Based on the above solution, we will let

$\stackrel{~}{g}(m, n)=\mathrm{IDFT}(H(k, l)F(k, l))$
Using this equation, we can calculate the value for each position on our final image, $\stackrel{~}{g}(m, n)$ . For example, due to the periodic extension of the images, when circular convolution is applied we willobserve a wrap-around effect.
$\stackrel{~}{g}(0, 0)=h(0, 0)f(0, 0)+h(1, 0)f(N-1, 0)+h(0, 1)f(0, N-1)+h(1, 1)f(N-1, N-1)$
Where the last three terms in are a result of the wrap-around effect caused by the presence of the images copies located all around it.

If the support of $h$ is $M$ x $M$ and $f$ is $N$ x $N$ , then we zero pad $f$ and $h$ to $M+N-1$ x $M+N-1$ (see ).

Circular Convolution = Regular Convolution

## Computing the 2d dft

$F(k, l)=\sum_{m=0}^{N-1} \sum_{n=0}^{N-1} f(m, n)e^{-i\frac{2\pi km}{N}}e^{-i\frac{2\pi ln}{N}}$
where in the above equation, $\sum_{n=0}^{N-1} f(m, n)e^{-i\frac{2\pi ln}{N}}$ is simply a 1D DFT over $n$ . This means that we will take the 1D FFT of each row; if wehave $N$ rows, then it will require $N\lg N$ operations per row. We can rewrite this as
$F(k, l)=\sum_{m=0}^{N-1} {f}^{}(m, l)e^{-i\frac{2\pi km}{N}}$
where now we take the 1D FFT of each column, which means that if we have $N$ columns, then it requires $N\lg N$ operations per column.
Therefore the overall complexity of a 2D FFT is $O(N^{2}\lg N)$ where $N^{2}$ equals the number of pixels in the image.

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Got questions? Join the online conversation and get instant answers!

#### Get Jobilize Job Search Mobile App in your pocket Now! By Brooke Delaney By Madison Christian By David Geltner By Stephen Voron By Savannah Parrish By Steve Gibbs By IES Portal By Brooke Delaney By OpenStax By Laurence Bailen